The physical implication of the gauge condition (REF ) is that one combination of the scalar fields is a gauge-artifact, and does not correspond to a propagating degree of freedom.The precise combination of Lagrangian fields which is physical and the corresponding form of its interactions depend on the gauge choice – although all physical amplitudes are gauge-invariant.
Note that the “mixing" of the scalar degree of freedom in the five-dimensional metric \(\hat{r}\)  with the bulk scalar field \(\hat{f}\)  only occurs in the presence of a \(y\) -dependent scalar background field configuration (\(\phi ^{\prime }_0\ne 0\) ). It is precisely this mixing between the two sectors that enables the dynamics which stabilize the size of the extra-dimension in the Goldberger-Wise mechanism [1]}, [2]} and which simultaneously give rise to a “radion" mass.
One advantage of working in this “unitary" gauge and eliminating the fluctuations of the scalar field \(\hat{f}\)  in favor of scalar fluctuations of the metric \(\hat{r}\)  is that all couplings linear in the physical scalar fields have the same algebraic form as couplings linear in the (massless) radion within the unstabilized model – simplifying the required coupling computations.
